Here is the first test in our game theory course. If you’re following along on this blog, you can do it and then look at the answers below.

### Definitions

**1.** Define a **Nash equilibrium** for a 2-player normal form

game.

**2.** Define the **expected value** of some function with respect to some probability distribution.

### Proof

**3.** Suppose and are the payoff matrices for 2-player normal form game. Prove that if is a Nash equilibrium, there cannot exist a choice for player A that strictly dominates choice

### 2-Player normal form games

Consider this 2-player normal form game:

**4.** Find all the Nash equilibria. Draw a box around each Nash

equilibrium.

**For problems 5-8 do not simplify your answers by working out the binomial coefficients, etc.**

### Probabilities

**5.** If you draw 3 cards from a well-shuffled standard deck,

what is the probability that at least 2 are hearts?

**6.** If you flip 4 fair and independent coins, what is the probability that exactly 2 land heads up?

### Expected values

**7.** Suppose you pay $2 to enter a lottery. Suppose you have a 1% chance of winning $100, and otherwise you win nothing. What is the expected value of your payoff, including your winnings but also the money you paid?

**8.** Suppose you draw two cards from a well-shuffled standard deck. Suppose you win $100 if you get two aces, $10 if you get one ace, and nothing if you get no aces. What is your expected payoff?

### Extra credit

About how many ways are there to choose 3 atoms from all the atoms in the observable universe? Since this question is for extra credit, I’ll make it hard: I’ll only accept answers written in **scientific notation**, for example

And here are the answers to the first test.

### Definitions

**1.** Given a 2-player normal form game where A’s

payoff is and B’s payoff is , a pair of choices is a **Nash equilibrium** if:

1) For all

2) For all

**2.** The **expected value** of a function with respect to a probability distribution on the finite set is

**Note that a good definition makes it clear what term is being defined, by writing it in boldface or underlining it. Also, it’s best if all variables used in the definition are explained: here they are and **

### Proof

**3. Theorem.** Suppose and are the payoff matrices for 2-player normal form game. If is a Nash equilibrium, there cannot exist a choice for player A that strictly dominates choice .

**Proof.** Suppose that is a Nash equilibrium. Then

for any choice for player A. On the other hand, if choice strictly dominates choice , then

This contradicts the previous inequality, so there cannot exist

a choice for player A that strictly dominates choice . █

Note that the really good way to write a proof involves:

**• First writing “Theorem” and stating the theorem.
• Saying “Proof” at the start of the proof.
• Giving an argument that starts with the hypotheses and leads to the conclusion.
• Marking the end of the proof with “Q.E.D.” or “█” or something similar.**

### 2-Player normal form games

**4.** In this 2-player normal form game, the three Nash equilibria are marked with boxes:

### Probabilities

**5.** If you draw 3 cards from a well-shuffled standard deck,

what is the probability that at least 2 are hearts?

**Answer.** One correct answer is

since there are:

• ways to choose 3 cards, all equally likely,

• ways to choose 2 hearts and ways to chose 1 non-heart, and

• ways to choose 2 hearts and ways to chose 0 non-hearts.

Another correct answer is

This is equal since and

**6.** If you flip 4 fair and independent coins, what is the probability that exactly 2 land heads up?

**Answer.** The probability

since there are possible ways the coins can land, all

equally likely, and ways to choose 2 of the 4 coins to land heads up.

### Expected values

**7.** Suppose you pay $2 to enter a lottery. Suppose you have a 1% chance of winning $100, and otherwise you win nothing. What is the expected value of your payoff, including your winnings but also the money you paid?

**Answer.** One correct answer is

Another correct answer is

Of course these are equal.

**8.** Suppose you draw two cards from a well-shuffled standard deck. Suppose you win $100 if you get two aces, $10 if you get one ace, and nothing if you get no aces. What is your expected payoff?

**Answer.** One correct answer is

since is the number of ways to pick aces and non-aces. Of course we can also leave off the last term, which is zero:

Since we can also write this as

Or, since and we can write this as

But I said not to bother simplifying the binomial coefficents.

### Extra credit

About how many ways are there to choose 3 atoms from all the atoms in the observable universe? Since this question is for extra credit, I’ll make it hard: I’ll only accept answers written in ** scientific notation**, for example

**Answer.** In class I said the number of atoms in the

observable universe is about and I said I might put this on the test. So, the answer is

Note that the question asked for an *approximate* answer, since we don’t know exactly how many atoms there are in the observable universe. The right answer to a question like this gives *no more decimal places than we have in our data*, so is actually *too precise!* You only have one digit in the data I gave you, so a better answer is

Since the figure is very approximate, another correct answer is